Trigonometry
Core ideas
Radians, arc length and sector area, the sine and cosine rules, trigonometric identities (Pythagorean, double-angle, sum-to-product), solving trigonometric equations in given intervals, and graphs of trig functions including amplitude, period, and phase shift. Inverse trig functions and their principal ranges also appear.
Traps
- Losing solutions when dividing by a trig factor; factorise instead of dividing, and check for zero-factor cases.
- Not accounting for the full interval: if the equation involves 2x, expand the interval to 2x first.
- Using the ambiguous case of the sine rule without checking both possible angles.
- Confusing degrees and radians; sector formulas (s = rθ, area = ½r²θ) require radians.
- Inverse sine returns values in [-π/2, π/2]; other solutions must be found from symmetry.
Rule of thumb
For equations like sin(kx) = c over an interval, substitute θ = kx, find all θ solutions over the scaled interval, then divide by k. This prevents missed solutions.
Worked example
Solve 2 sin^2 x - sin x - 1 = 0 for 0 ≤ x < 2π. Factorise: (2 sin x + 1)(sin x - 1) = 0. So sin x = -1/2 or sin x = 1. For sin x = -1/2, x = 7π/6 or 11π/6. For sin x = 1, x = π/2. The solutions are x = π/2, 7π/6, 11π/6.